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Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components)

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  • Blank, Aharon
  • Solomon, Sorin
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    Abstract

    We study a few dynamical systems composed of many components whose sizes evolve according to multiplicative stochastic rules. We compare them with respect to the emergence of power laws in the size distribution of their components. We show that the details specifying and enforcing the smallest size of the components are crucial as well as the rules for creating new components. In particular, a growing system with a fixed number of components and a fixed smallest component size does not converge to a power law. We present a new model with variable number of components that converges to a power law for a very wide range of parameters. In a very large subset of this range, one obtains for the exponent α the special value 1 specific for the city populations distribution. We discuss the conditions in which α can take different values. In the case of the stock market, the distribution of the investors’ wealth is related to the ratio between the new capital invested in stock and the rate of increase of the stock index.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0378437100004647
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    Bibliographic Info

    Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

    Volume (Year): 287 (2000)
    Issue (Month): 1 ()
    Pages: 279-288

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    Handle: RePEc:eee:phsmap:v:287:y:2000:i:1:p:279-288

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    Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

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    Cited by:
    1. Esteban Rossi-Hansberg & Mark L.J. Wright, 2005. "Urban Structure and Growth," NBER Working Papers 11262, National Bureau of Economic Research, Inc.
    2. repec:ebl:ecbull:v:15:y:2003:i:6:p:1-7 is not listed on IDEAS
    3. Itzhack, Royi & Muchnik, Lev & Erez, Tom & Tsaban, Lea & Goldenberg, Jacob & Solomon, Sorin & Louzoun, Yoram, 2010. "Empirical extraction of mechanisms underlying real world network generation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(22), pages 5308-5318.
    4. Bucsa, G. & Jovanovic, F. & Schinckus, C., 2011. "A unified model for price return distributions used in econophysics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(20), pages 3435-3443.
    5. Y. Malevergne & A. Saichev & D. Sornette, 2010. "Zipf's law and maximum sustainable growth," Papers 1012.0199, arXiv.org.
    6. Jan Eeckhout, 2004. "Gibrat's Law for (All) Cities," American Economic Review, American Economic Association, vol. 94(5), pages 1429-1451, December.
    7. V. Gontis, 2002. "Multiplicative Stochastic Model of the Time Interval between Trades in Financial Markets," Papers cond-mat/0211317, arXiv.org.
    8. Segarra, Agustí & Teruel, Mercedes, 2012. "An appraisal of firm size distribution: Does sample size matter?," Journal of Economic Behavior & Organization, Elsevier, vol. 82(1), pages 314-328.
    9. Corrado Di Guilmi & Mauro Gallegati & Edoardo Gaffeo, 2003. "Power Law Scaling in the World Income Distribution," Economics Bulletin, AccessEcon, vol. 15(6), pages 1-7.
    10. Guido Fioretti, 2005. "The Production Function," Papers physics/0511191, arXiv.org.
    11. Gerrit de Wit, 2004. "Firm Size Distributions : An overview of steady-state distributions resulting from firm dynamics models," Scales Research Reports N200418, EIM Business and Policy Research.
    12. Fioretti, Guido, 2007. "The production function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 707-714.
    13. Navarro-Barrientos, Jesús Emeterio & Cantero-Álvarez, Rubén & Matias Rodrigues, João F. & Schweitzer, Frank, 2008. "Investments in random environments," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(8), pages 2035-2046.
    14. Andrea Bonaccorsi & Maurizio Martinelli & Cristina Rossi & Irma Serrecchia, 2002. "Measuring and modelling Internet diffusion using second level domains: the case of Italy," LEM Papers Series 2002/17, Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy.

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